Remarks on Parallel Analysis.

We investigate parallel analysis (PA), a selection rule for the number-of-factors problem, from the point of view of permutation assessment. The idea of applying permutation test ideas to PA leads to a quasi-inferential, non-parametric version of PA which accounts not only for finite-sample bias but sampling variability as well. We give evidence, however, that quasi-inferential PA based on normal random variates (as opposed to data permutations) is surprisingly independent of distributional assumptions, and enjoys therefore certain non- parametric properties as well. This is a justification for providing tables for quasi-inferential PA. Based on permutation theory, we compare PA of principal components with PA of principal factor analysis and show that PA of principal factors may tend to select too many factors. We also apply parallel analysis to so-called resistant correlations and give evidence that this yields a slightly more conservative factor selection method. Finally, we apply PA to loadings and show how this provides benchmark values for loadings which are sensitive to the number of variables, number of subjects, and order of factors. These values therefore improve on conventional fixed thresholds such as 0.5 or 0.8 which are used irrespective of the size of the data.